I am reading a text where given an expression for omitted variable bias $$(1) \ \hat \lambda = \lambda + (D^\top M_XD)^{-1} D^\top M_X F\psi$$ it is stated that if $D$ is orthogonal to $X$ such that $(D^\top M_XD) = D^\top D$ and $X$ is orthogonal to $F$ such that $ D^\top M_X F = D^\top F$ then
$$(2) \ \hat \lambda = \lambda + (D^\top D)^{-1} D^\top F\psi$$
my question is why is it necessary for $X$ to be orthogonal to $F$ to get that result?
Couldn't I just reason that if $X$ is orthogonal to $D$ then $M_X D = D$ and $D^\top M_X^\top =D^\top M_X = D^\top$ and simply use these identities to get from (1) to (2)? Problem is I do not know what is meant by orthogonality in this context and whether these identities follow from however orthogonality is defined.
The definition of $M_X = I - X(X^\top X)^{-1}X^\top$
[Note: The author does not give an explicit definition of orthogonality]